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94 V NORMED SPACES

(/„) is a Cauchy sequence which does not converge (if there existed a limit g of (/„) in E,
show that one would necessarily have g(t) = 1 for 0 ^ / ^ i and #(0 = 0 for J < t ^ 1,
which would violate the continuity of g).

(b) Show that the distance on E defined in (5.1.4) is not topologically equivalent to the
distance defined in (3.2.4). (Give an example of a sequence in E which tends to 0 for
\\f-ff\\i, but has no limit for the distance defined in (3.2.4).)

2. If A, B are two subsets of a normed space E, we denote by A -f- B the set of all sums
a 4- 6, where a e A, b e B.

(a) Show that if one of the sets A, B is open, A + B is open.

(b) Show that if both A and B are compact, A + B is compact (use (3.17.9) and
(3.20.16)).

(d) Give an example of two closed subsets A, B of R such that A + B is not closed
(cf. the example given before (3.4.1)).

3. Let E be a normed space.

(a) Show that in E the closure of an open ball is the closed ball of same center and
same radius, the interior of a closed ball is the open ball of same center and same
radius, and the frontier of an open ball (or of a closed ball) is the sphere of same center
and same radius (compare to Section 3.8, Problem 4).

(b) Show that the open ball B(0; r) is homeomorphic to E (consider the mapping
_x-*rxl(l+\\x\\)).

4. In a normed space E, a segment is the image of the interval [0,1 ] of R by the continuous
mapping / -> ta + (1 — t)b, where a e E and b e E; a and b are called the extremities of
the segment. A segment is compact and connected. A broken line in E is a subset L of E
such that there exists a finite sequence (jtf)o^ i«« of points of E having the property that
if S_f is the segment of extremities x_t and x_t+1 for 0 *£ / ^ n — 1, L is the union of the S₄;
the sequence (x_t) is said to define the broken line L (a given broken line may be defined
in general by infinitely many finite sequences). If A is a subset of E, a, b two points of A,
one says that a and b are linked by a broken line in A, if there is a sequence (xi)o*i*n
such that a = x₀, b = x_n and that the broken line L defined by that sequence is con-
tained in A.

If any two points of A can be linked by a broken line in A, A is connected. Conversely,
if A c E is a connected open set, show that any two points of A can be linked by a
broken line in A (prove that the set of points y e A which can be linked to a given point
a e A by a broken line in A is both open and closed in A).

5. In a real vector space E, a linear variety V is a set of the form a + M, where M is a
linear subspace of E; the dimension (resp. codimension) of V is by definition the dimen-
sion (resp. codimension) of M. If b $ V and if V has finite dimension p (resp. finite
codimension q)₉ the smallest linear variety W containing both b and V has finite dimen-
sion^ + 1 (resp. finite codimension g — 1).

Let A be an open connected subset of a real normed space E, and let (V_n) be a denumer-
able sequence of linear varieties in E, each of which has codimension ^ 2; show that
if B is the union of the V_fl, A n (E — B) is connected. (Hint: Use Problem 4; if L
is a broken line linking two points a, b of A n (E — B) in A, prove that there exists an-
other broken line L' "close" to L, contained in A n (E — B). To do that, observe that
if x e E — B, the set of points y e E such that the segment of extremities x, y does not
meet any V_n, is dense in E, using (2.2.17).)

In particular, if the dimension of E is 5*2, and if D is a denumerable subset of E,
A n (E — D) is connected.



	94 V NORMED SPACES (/„) is a Cauchy sequence which does not converge (if there existed a limit g of (/„) in E, show that one would necessarily have g(t) = 1 for 0 ^ / ^ i and #(0 = 0 for J < t ^ 1, which would violate the continuity of g). (b) Show that the distance on E defined in (5.1.4) is not topologically equivalent to the distance defined in (3.2.4). (Give an example of a sequence in E which tends to 0 for \\f-ff\\i, but has no limit for the distance defined in (3.2.4).) 2. If A, B are two subsets of a normed space E, we denote by A -f- B the set of all sums a 4- 6, where a e A, b e B. (a) Show that if one of the sets A, B is open, A + B is open. (b) Show that if both A and B are compact, A + B is compact (use (3.17.9) and (3.20.16)). (c) Show that if A is compact and B is closed, then A 4- B is closed. (d) Give an example of two closed subsets A, B of R such that A + B is not closed (cf. the example given before (3.4.1)). 3. Let E be a normed space. (a) Show that in E the closure of an open ball is the closed ball of same center and same radius, the interior of a closed ball is the open ball of same center and same radius, and the frontier of an open ball (or of a closed ball) is the sphere of same center and same radius (compare to Section 3.8, Problem 4). (b) Show that the open ball B(0; r) is homeomorphic to E (consider the mapping _x-rxl(l+\\x\\)).* 4. In a normed space E, a segment is the image of the interval [0,1 ] of R by the continuous mapping / -> ta + (1 — t)b, where a e E and b e E; a and b are called the extremities of the segment. A segment is compact and connected. A broken line in E is a subset L of E such that there exists a finite sequence (jtf)o^ i«« of points of E having the property that if S_f is the segment of extremities x_t and x_t+1 for 0 £ / ^ n —* 1, L is the union of the S₄; the sequence (x_t) is said to define the broken line L (a given broken line may be defined in general by infinitely many finite sequences). If A is a subset of E, a, b two points of A, one says that a and b are linked by a broken line in A, if there is a sequence (xi)oin such that a = x₀, b = x_n and that the broken line L defined by that sequence is con- tained in A. If any two points of A can be linked by a broken line in A, A is connected. Conversely, if A c E is a connected open set, show that any two points of A can be linked by a broken line in A (prove that the set of points y e A which can be linked to a given point a e A by a broken line in A is both open and closed in A). 5. In a real vector space E, a linear variety V is a set of the form a + M, where M is a linear subspace of E; the dimension (resp. codimension) of V is by definition the dimen- sion (resp. codimension) of M. If b $ V and if V has finite dimension p (resp. finite codimension q)₉ the smallest linear variety W containing both b and V has finite dimen- sion^ + 1 (resp. finite codimension g — 1). Let A be an open connected subset of a real normed space E, and let (V_n) be a denumer- able sequence of linear varieties in E, each of which has codimension ^ 2; show that if B is the union of the V_fl, A n (E — B) is connected. (Hint: Use Problem 4; if L is a broken line linking two points a, b of A n (E — B) in A, prove that there exists an- other broken line L' "close" to L, contained in A n (E — B). To do that, observe that if x e E — B, the set of points y e E such that the segment of extremities x, y does not meet any V_n, is dense in E, using (2.2.17).) In particular, if the dimension of E is 5*2, and if D is a denumerable subset of E, A n (E — D) is connected.